Section 10.4 Vectors

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1 220 Section 10.4 Vectors In this section, we will define and explore the properties of vectors. Vectors can be used to represent the speed and the direction of an object, the force and direction acting on an object, and many other physical applications. A vector has two parts: its magnitude and direction. Thus, if a plane is traveling 180 mph with a bearing of S35 E, we can represent the speed and the direction as a velocity vector. Generally, we use directed line segments to graphically represent a vector. The initial point is where the vector starts and the arrow of the ray points to the terminal point where the vector ends. We will use boldface lowercase letter or we will use the uppercase letters of the initial and terminal points with an arrow above the two letters to represent a vector. Q PQ or v or v P v or PQ denotes the magnitude of the vector. If a vector has a magnitude of 0, then it is called the zero vector, 0 or 0. Two vectors are equal if and only if they have the same magnitude and the same direction. Let u, v, w be vectors. Addition of vectors have several properties that it satisfies: Commutative Property v + w = w + v Associative Property u + (v + w) = (u + v) + w Identity Property Inverse Property v + 0 = v v + ( v) = 0 Note: v w = v + ( w)

2 A scalar is simply a real number without direction. In other words, it has magnitude, but no direction. If we let v be a vector and α be a scalar, then we can define scalar multiplication as the following: Definition: Scalar Multiplication Case 1: If α > 0 and v 0, then αv is a vector with magnitude of α v in the same direction as v. Case 2: If α < 0 and v 0, then αv is a vector with magnitude of α v in the opposite direction of v. Case 3: If α = 0 or v = 0, then αv = 0. Scalar Multiplication has the following properties: 0 v = 0 1 v = v 1 v = v (α + β)v = αv + βv α(v + w) = αv + αw α(βv) = (αβ)v 221 v v 2v Objective 1: Graph Vectors. When adding or subtracting vectors, we draw the first vector and then draw the next such that the initial point of the second vector is the terminal point of the first vector. Use the vectors below to find the following: u v w Ex. 1a u 2w Ex. 1b 3w + 2v Ex. 1c 2u + v + 2w

3 a) First draw u and then draw 2w such that the initial point of 2w is the terminal point of u. 222 u 2w u 2w b) First draw 3w and then draw + 2v such that the initial point of 2v is the terminal point of 3w. 3w 2v 3w + 2v c) First draw 2u and then draw v such that the initial point of v is the terminal point of 2u. Finally, draw 2w such that the initial point of 2w is the terminal point of v. 2u v 2w 2u + v + 2w

4 223 The magnitude of v, denoted v, is the length of the directed line segment. The magnitude has the following properties: v Theorem Let v be a vector and α be a scalar, then 1) v 0 2) v = 0 if and only if v = 0. 3) v = v 4) αv = α v Definition A vector u is called a unit vector if u = 1. Objective #2: Find a Position Vector. We can think of a vector has being the hypotenuse of a right triangle. The hypotenuse of a right triangle is determined by the two legs a and b. Thus, using a and b, we can then represent a vector algebraically: Definition An algebraic vector v is represented as v = a, b where a and b are scalars and are called the components of v.if the initial point of v is at the origin and the teriminal point is (a, b), then v = a, b is called a position vector. A position vector always starts at the origin. Theorem If v is a vector with an initial point of (x 1, y 1 ) and a terminal point of (x 2, y 2 ), then v is equal to the position vector x 2 x 1, y 2 y 1. (x 2, y 2 ) (x 1, y 1 ) (x 2 x 1 ) Find the position vector: (y 2 y 1 ) Ex. 2 v = P 1 P 2 where P 1 = ( 3, 4) and P 2 = (5, 2). v = 5 ( 3), 2 4 = 8, 2 We can say that two vectors are equal if they have the same position vector. This leads to the following theorem:

5 224 Equality of Vectors Theorem Two vectors are equal if and only if their components are equal. In other words, if v = a 1, b 1 and w = a 2, b 2, then v = w if and only if a 1 = a 2 and b 1 = b 2. We now want to be able to represent vectors another way so that it will be easier to combine them algebraically: Definition Component Unit Vectors Let i = 1, 0 be the unit vector with direction along the positive x- axis. Let j = 0, 1 be the unit vector with direction along the positive y- axis. Using this definition, we can write any vector using the unit vectors i and j. Thus, we can say that v = a, b = a 1, 0 + b 0, 1 = ai + bj, where a and b are called the horizontal and vertical components of v respectively. Objective #3 & #4: Working with Vectors Algebraically. Definition Let v = a 1, b 1 = a 1 i + b 1 j and w = a 2, b 2 = a 2 i + b 2 j be two vectors and let α be a scalar. Then 1) v + w = (a 1 + a 2 )i + (b 1 + b 2 )j = a 1 + a 2, b 1 + b 2 2) v w = (a 1 a 2 )i + (b 1 b 2 )j = a 1 a 2, b 1 b 2 3) αv = (αa 1 )i + (αb 1 )j 4) v = a 1 2 +b 1 2 If v = 3i 4j and w = 2i + j, find the following: Ex. 3a v + w Ex. 3b v w Ex. 3c 4v 2w Ex. 3d v a) Think of combining like terms: v + w = (3i 4j) + ( 2i + j) = i 3j b) Think of combining like terms: v w = (3i 4j) ( 2i + j) = 3i 4j + 2i j = 5i 5j c) Distribute and combine like terms: 4v 2w = 4(3i 4j) 2( 2i + j) = 12i 16j + 4i 2j = 16i 18j d) v = (3) 2 +( 4) 2 = 25 = 5

6 225 Objective #5: Finding Unit Vectors. In some instances, it is useful to find a unit vector in the same direction as a vector v. Theorem Unit Vector in the Direction of v Given a nonzero vector v, the unit vector in the same direction as v is u = v v This also implies that v = v u. We will see where this relationship is important in the next few pages. Find a unit vector in the same direction as v: Ex. 4a v = 8i + 6j Ex. 4b v = 3i j a) First, find v : v = ( 8) 2 +(6) 2 = 100 = 10 Now, u = v v = 8i+6j 10 b) First, find v : = 0.8i + 0.6j v = (3) 2 +( 1) 2 = 10 Now, u = v v = 3i j = 3 i 1 j = i j Objective #6: Find a Vector from its Direction and Magnitude. There are many vectors that are describe in terms of its magnitude and direction. For instance, a velocity vector is describe in terms of its magnitude (speed) and direction. An object may be moving at 35 miles per hour at an angle of evaluation of 15. If we think of the magnitude v as a radius of a circle centered at the origin and the direction as an angle α from the positive x-axis or from i, then v = ai + bj can be expressed as: v = v cos(α)i + v sin(α)j = v (cos(α)i + sin(α)j) where 0 α < 360

7 226 Find the vector that satisfies the following: Ex. 5 v = 3 and α = 255 v = v (cos(α)i + sin(α)j) = 3(cos(255 )i + sin(255 )j) = 3cos(255 )i + 3sin(255 )j i j Write v in the form v (cos(α)i + sin(α)j): Ex. 6 v = 1.5i 2j Since a = 1.5 and b = 2, the terminal side is in quadrant IV First, calculate v : v = (1.5) 2 + ( 2) 2 = 6.25 = 2.5 Now, find α: tan(α) = α = tan 1 ( ) = But α needs to be in [0, 360 ), so we will add 360 to our answer: Thus, v = 2.5(cos( )i + sin( )j) Objective #7: Model with Vectors Since forces can be represented as vectors, we can combine two forces to represent a resultant force. In a system, the resultant force on an object is 0 and the object is at rest, then the object is in static equilibrium. Solve the following: Ex. 7 If a child pushes a cart with a force of 10 pounds up 20 incline, express the force vector in terms of i and j. F = F (cos(α)i + sin(α)j) = 10(cos(20 )i + sin(20 )j) 9.397i j Ex. 8 Suppose an object is suspended by two ropes as illustrated below. If the object weighs 600 pounds, find the tension on each rope

8 227 There are three forces i this system. Let F 1 be the tension on the rope on the right side, F 2 be the tension on the rope on the left side, and F 3 be the force due to gravity. Since alternate interior angles of two parallel lines cut by a transversal are equal, then angle between the rope on the right and the positive x-axis is 48. For the rope on the left, the alternate interior angle is 37, so the angle between the positive x-axis and that rope is = Thus, for the two ropes: F 1 = F 1 (cos(48 )i + F 1 (sin(48 )j F 2 = F 2 (cos(143 )i + F 2 (sin(143 )j For the weight of the object, gravity pulls it in the negative y-axis direction, so F 3 = 600j Since the object is in static equilibruim, the resultant force is 0. F r = F 1 + F 2 + F 3 = a r i + b r j = 0 which implies a r and b r are zero which means that the sum of each components will be zero: i component: F 1 (cos(48 ) + F 2 (cos(143 ) = 0 j component: F 1 (sin(48 ) + F 2 (sin(143 ) 600 = 0 Looking at the first equation, we can solve for F 1 : F 1 (cos(48 ) + F 2 (cos(143 ) = 0 F 1 (cos(48 ) = F 2 (cos(143 ) F 1 = F 2 (cos(143 )/(cos(48 ) = F 2 Now, substitute in for F 1 in the j component equation: F 1 (sin(48 ) + F 2 (sin(143 ) 600 = F 2 (sin(48 ) + F 2 (sin(143 ) 600 = 0 Now, solve for F 2 : F 2 (sin(48 ) + F 2 (sin(143 ) 600 = F 2 ( ) + F 2 ( ) 600 = F 2 + F 2 ( ) 600 = F = F 2 = 600 F Now, find F 1 :

9 F 1 = F 2 = ( ) The rope on the right has a tension of pounds and the rope on the left has a tension of pounds. Ex. 9 Rowing across a lake, Roger maintains a constant speed of 8 miles per hour due north. If the current is 3 miles per hour in a southeasterly direction, find the actual speed & direction of the boat. Let v b = the velocity of the boat v c = the velocity of the current v n = the net velocity (what we are trying to find) We need to diagram the situation so we can write down our vectors: Since v b lies on the positive y-axis, we can represent the vector as v b = 0i + 8j. v b The vector v c has a southeasterly direction so the angle v n from the positive x-axis to that vector is 45. Since the magnitude of the vector is 3, then v c v c = 3cos( 45 )i + 3sin( 45 )j = i j. v n = v b + v c = 8j i j = i + ( )j. The actual speed of the boat is v n = (1.5 2 ) 2 + ( ) 2 = = = mph To get the direction, we need to find the angle between v b and v n. First, we will find the angle between v n and the x-axis: tan(θ) = y x = θ = tan 1 ( ) = Thus, the angle between v b and v n is The boat is moving at 6.25 mph with a bearing of N19.84 E. 228

SECTION 6.3: VECTORS IN THE PLANE

(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,